Harmonic motion is one of the basic type of motion in nautre. Plenty of bodies and items execute harmonic motion during their normal existence. Every motion is described by motion equation. Motion equation is differential equation.
Solution of motion equation for simple harmonic oscillator. Harmonic oscillator is a spring with mass hanged at its end. Period of harmonic vibrations and expression for road, velocity and acceleration in function of time will we calculated.
Simple harmonic oscillator
Solution of motion equation for suppressed harmonic oscillator. Suppressed harmonic oscillator is a spring with mass hanged at its end. Spring of suppressed harmonic oscillator has beside stiffness also suppression ability.
Suppressed harmonic oscillator
Mathematical pendulum is built with massless rope with length l. One of rope’s ends is fixed to the ceiling. To the second end of rope is fixed to the material point with mass m. Material point is then hanged to the rope. When mathematical pendulum is in equilibrium material point is not moving. In equilibrium position gravity force is balanced by rope’s tension force. In a certain moment mathematical pendulum was deflected from its equilibrium and was inclined from vertical position by angle α. Mathematical pendulum is under gravity field, so gravity force works on it.
Mathematical pendulum - motion equation
Physical pendulum is built with rigid body. One of rigid body ends is fixed to the ceiling. Rigid body is able to rotate around axis which is placed exactly in place where rigid’s end is fixed. Rotation axis is perpendicular to the plane of drawing. Rigid body has mass m and inertia I. Rigid body length is 2∙l. Note that inertia I is known for axis of rotations. If physical pendulum is in equilibrium position then it is not moving. In equilibrium position gravity force is balanced by rigid body’s reaction force. In a certain moment pendulum was deflected from its equilibrium and was inclined from vertical position by angle α. Pendulum is under gravity field, so gravity force works on it. Remember that physical pendulum is rigid body so gravity force is placed on rigid body’s gravity center C.
Physical pendulum - motion equation
On movable rolls lies board. Board's mass is m1. Board's length is l. Rollers has equal mass m2. Rollers execute rotary motion which rotational velocity is ω. Rollers are rotating in opposite directions. Between board's surface and rolls is kinetic sliding friction which is described by sliding friction factor ƒ.
Harmonic vibrations - example 1
Inside U-tube is fluid. Mass of fluid inside tube is m. Fluid density is ρ. Area tube's section is S. Both ends of U-tube are open. In one moment fluid was moved from its equilibrium position by distance Δx.
Harmonic vibrations - example 2
oller is submerged in water. Roller has mass m. Density of water is known and its value is ρ. When roller is in equilibrium position it is submerged in water on depth x. Two forces work on roller: gravity force and buoyancy force.
Harmonic vibrations - example 3